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A circle is inscribed in an equilateral triangle. Smaller circles touch it and the sides of the triangle, the process continuing indefinitely. What is the sum of the areas of all the circles?

### Climbing Powers

$2\wedge 3\wedge 4$ could be $(2^3)^4$ or $2^{(3^4)}$. Does it make any difference? For both definitions, which is bigger: $r\wedge r\wedge r\wedge r\dots$ where the powers of $r$ go on for ever, or $(r^r)^r$, where $r$ is $\sqrt{2}$?

A quadrilateral changes shape with the edge lengths constant. Show the scalar product of the diagonals is constant. If the diagonals are perpendicular in one position are they always perpendicular?

# Weekly Challenge 10: Solve Me!

##### Stage: 5 Short Challenge Level:

Find a solution to this equation to 1 dp.

$$2x^3+34 x^2+567x +8901=0$$

Are there any others?

Did you know ... ?

Numerical solution of equations forms an important part of real-world mathematics and mathematics applied to science, where equations are often too complex to be solved exactly. Mathematicians have developed many advanced techniques for the numerical solution and exploration of equations.